There are basically two main areas within decision theory: probabilistic decisions and multi-criteria decisions. Each of these contains theories, methods, and procedures for aiding decision makers. A decision maker can be, i.a., one human, a group of humans, a computer program, or a computer software controlled machine. Decision models can be grouped into different types that can be termed Probabilistic Decision models and Multi-Criteria Decision models, see:
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Probabilistic Decision models are often given a tree representation. Consider the tree in FIG. 6. The decision tree consists of a root, representing a decision, a set of intermediary (event) nodes, representing some kind of uncertainty and consequence nodes, representing possible final outcomes. Usually probability distributions are assigned as weights in the probability nodes as measures of the uncertainties involved. The informal semantics are simply that given that an alternative Ai is chosen there is a probability pij that an event Hij occurs. This event can be a consequence with a value vijk assigned to it or another event. Usually, the maximization of the expected value is used as an evaluation rule. For instance, in FIG. 6, the expected value of alternative Ai is:
      E    ⁡          (              A        i            )        =            ∑              j        =        1            2        ⁢                  p        ij            ⁢                        ∑                      k            =            1                    2                ⁢                              p            ijk                    ⁢                                    v                              ijk                ⁢                                                                  ⁢                1                                      .                              
There are also several approaches for multi-criteria decision making, where the decision criteria can be arranged in hierarchies. See FIG. 7. Examples include the analytical hierarchy process (AHP), the Evidential Reasoning (ER) approach, and various methods by Salo et al above.
On each level, the criteria are given weights and the alternatives are valued with respect to each sub-criterion. The maximization of the weighted value is usually used as an evaluation rule. For instance, in FIG. 7, the value of alternative Ai under sub-criterion jk is denoted vijk. The weight of criteria j is denoted by wj. Then the weighted value of alternative Ai is:
      E    ⁡          (              A        i            )        =            ∑              j        =        1            2        ⁢                  w        j            ⁢                        ∑                      k            =            1                    2                ⁢                              w            jk                    ⁢                                    v              ijk                        .                              
Since the expected value is a weighting operation, both of the above approaches, probabilistic decision models and multi-criteria decision models, selects the alternative with the greatest weighted value. However, no combination rules or algorithms for the evaluation of such a combination have been proposed. Note that in this presentation, the concepts “expected value” and “weighted value” are used interchangeably for the expected value in probabilistic models as above, for the weighted value in multi-criteria models as above, and for the combined generalized expected value containing both criteria weights and probabilities as introduced below.
Risk Analysis is the task of determining the risk involved with a particular action, often resulting in a chain of events, each event path through the risk tree ending in a final consequence. Risk analyses are often displayed using a tree representation. The risk tree consists of a set of intermediary (event) nodes, representing some kind of uncertainty and consequence nodes, representing possible final outcomes. The kinship with probabilistic decisions is strong. A risk analysis can be equally regarded as a decision between a main alternative (the risk situation) and a zero alternative with only one final consequence having the value zero. Thus, when referring to decision problems, the methods and software and tools as described herein can equally be applied to risk analysis.
Probabilistic networks contain chains of events dependent on each other, where a prior node is a precondition for the subsequent node in the (directed) path. While not specifically treated in the sequel, the multiplication of probabilities occurring in probabilistic networks is computationally analogous to the evaluation of event chains in decision trees. Furthermore, decisions made in probabilistic networks through conversion into influence diagrams are solved using the same tree methods and procedures presented here. Thus, when referring to decision problems, the methods and software and tools as described herein can equally be applied to other decision models such as probabilistic networks and influence diagrams.
The information available in decision making is often incomplete, vague, and imprecise, and several decision models have been suggested to handle such situations. During the last 50 years, various methods based on interval estimates of probabilities and values of any sort not only numerical values have been suggested. Even if these approaches generally are well-founded, much less has been done to take the evaluation perspective into consideration and, in particular, computational aspects and implementational issues.
A number of models with representations allowing imprecise probability statements have been suggested over the years. Some of them are based on capacities, evidence theory and belief functions, various kinds of logic, upper and lower probabilities, or sets of probability measures. The common characteristic of the approaches is that they typically do not include the additivity axiom of probability theory and consequently do not require a decision maker to model and evaluate a decision situation using precise probability (and, in some cases, value) estimates. For some overviews, see, e.g., [Weichselberger & Pöhlman, 1990], [Walley, 1991], and [Ekenberg & Thorbiörnson, 2001].
These have been more concerned with representation and less with evaluation. Moreover, very few have addressed the problems of computational complexity when solving decision problems involving interval estimates. It is important to be able to determine, in a reasonably short time, how various evaluative principles rank the given options in a decision situation.
Interval approaches have also been considered in order to extend decision models for multi-criteria decision making. The method PRIME in [Salo & Hämäläinen, 2001] is a generalization of value tree analysis. Similarly, the preference programming method [Salo & Hämäläinen, 1995] extends the analytical hierarchy process (AHP). These approaches are limited in several respects with respect to expressibility and evaluation capabilities. Neither do they address probabilities or consequence structures. [Park & Kim, 1997] is an attempt to combine criteria weights with probabilistic reasoning, treating only one-level criteria, and can only rank the options without cardinal aspects. A one-level representation in this manner, but with larger expressibility compared to [Park & Kim, 1997], is provided in [Danielson et al, 2003a] and [Ekenberg et al, 2003].
Further, some approaches for extending the representation using distributions over classes of probability and utility measures have been proposed. These have been developed into various hierarchical models, such as second-order probability theory [Gärdenfors & Sahlin, 1982], [Ekenberg & Thorbiörnson, 2001]. The former consider belief distributions, but restricted to the probability case and interval representations. Another limitation is that it does not address the relation between distributions over spaces of one or several dimensions respectively. The same criticism applies to [Hodges & Lehmann, 1952], [Hurwicz, 1951], and [Wald, 1950]. None of these, nor [Ekenberg & Thorbiörnson, 2001], handle the issues of tree representation or evaluation at all. Furthermore, no detailed procedures or suggestions are provided for how to represent or how to evaluate aggregations of belief distributions.
The Delta method for handling vague and imprecise information has been developed in a number of papers, e.g. [Danielson & Ekenberg, 1998] and [Ekenberg, 2000]. The method has been used in a wide variety of contexts, e.g., deposition of nuclear waste, investment situations, and evaluation of offers in purchase situations [Danielson et al, 2003b]. The method was invented to counter the problems with unnatural precision and to provide computational concepts for handling imprecise probabilities and values. The approach is a single-level approach, not able to handle multi-level decision trees where the outcome of one event can depend on previous outcomes. The Delta method as it is known to date does neither use general multi-criteria hierarchies nor algorithms for trees. Earlier known algorithms are not applicable to problems with trees, be they probabilistic, multi-criteria, or a combination thereof. Neither belief distributions nor procedures for aggregated interval estimates (such as weighted expected values) have been considered earlier.